Mathematical Aspects of Vacuum Energy on Quantum Graphs

Abstract

We use quantum graphs as a model to study various mathematical aspects of the vacuum energy, such as convergence of periodic path expansions, consistency among different methods (trace formulae versus method of images) and the possible connection with the underlying classical dynamics. We derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph. For an important special case of graphs with equal bond lengths, we derive a simpler explicit formula. The main results are derived using the trace formula. We also discuss an alternative approach using the method of images and prove that the results are consistent. This may have important consequences for other systems, since the method of images, unlike the trace formula, includes a sum over special ``bounce paths''. We succeed in showing that in our model bounce paths do not contribute to the vacuum energy. Finally, we discuss the proposed possible link between the magnitude of the vacuum energy and the type (chaotic vs. integrable) of the underlying classical dynamics. Within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy.

Fixed several typos, explain the use of random matrices in Section 6

References in corpus (18)

• New Developments in the Casimir Effect
• The Casimir effect: Recent controversies and progress
• Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Kirchhoff's Rule for Quantum Wires
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems
• Casimir force on a piston
• The Casimir Effect and Geometric Optics
• Casimir force between partially transmitting mirrors
• Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers
• Vacuum Energy and Repulsive Casimir Forces in Quantum Star Graphs
• Repulsive Casimir Pistons
• The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
• Unitary stochastic matrix ensembles and spectral statistics
• Spectral statistics for unitary transfer matrices of binary graphs
• Quantum Fields on Star Graphs
• Index theorems for quantum graphs
• Quantum graphs where back-scattering is prohibited

Cited by in corpus (5)

• Non-equilibrium Steady States of Quantum Systems on Star Graphs
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Zeta functions of quantum graphs
• Relating Zeta Functions of Discrete and Quantum Graphs
• Vacuum energy of Schrödinger operators on metric graphs